课程名称︰分析导论优一
课程性质︰数学系大二必修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/10/21
考试时限（分钟）：190
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
1. (a) (10%) Prove that Cantor's Intersection Theorem holds for nested compact
sets in an arbitrary metric space, namely, if K , K , ... is a nested
1 2
sequence of nonempty compact sets in a metric space S, then
∞
∩ K ≠ψ
j=1 j
(b) (10%) In (a), can we replace "compact sets" by "closed and bounded
sets"?
n n 2
2. Let S = {x = (x , x , ..., x ) ∈ |R (n > 1) : Σ x = 1}. Assume that f
1 2 n k=1 k
is continuous function on S.
(a) (10%) If f(x)≠0 for all x∈S and f(x ) > 0 for some x ∈S. Show that
0 0
there exists a constant c > 0 such that f(x) ≧ c for all x∈S.
(b) (10%) Show that there exists a pair of diametrically opposite points on
S at which f assumes the same value, i.e., for some x∈S, f(x) = f(-x).
3. (a) (10%) Let f: S → T be a continuous function. Assume that S is compact.
Show that for any Cauchy sequence {x } in S, {f(x )} is a Cauchy
n n
sequence in T.
(b) (10%) Does the result in (a) remain true if we remove the compactness
assumption on S?
4. (20%) Let f be a real-valued function defined on [a, b]. The graph of f on
[a, b] is given by
2
G(f) = {(x, f(x))∈|R : x∈[a, b]}
Show that f is continuous on [a, b] iff G(f) is closed and connected in
2
|R .
5. (a) (10%) Let (M, d) be a metric space, define
d(x, y)
d'(x, y) =──────
1 + d(x, y)
Prove that d' is also a metric for M.
(b) (10%) Is any open set in (M, d) open in (M, d') and vice versa?